Conductance quantum
Encyclopedia
The conductance quantum is the quantized unit of conductance. It is defined as G0 = 2e
2/h
= ≈ Ω−1. It appears when measuring the conductance of a quantum point contact
.
The name conductance quantum is somewhat misleading, since it implies that there can only be conductances that are integer multiples of G0. This is not the case, instead conductance quantum means the conductance of one conductance channel, if the probability of an electron entering this channel being transmitted is unity. If this probability is not unity, there must be a correction for the particular conductance channel.
An Intuitive Derivation from "uncertainty principle
The Quantum of Conductance can easily be derived and seen to be as a consequence of the time-energy uncertainty.
Let us assume that a wire is carrying a current I. The current is given by e
/τ, where τ is transit time and the e is electronic charge. When a potential of V is applied, then,
E=e
V and I= e
/τ, which implies τ=e
/I,
then, (e
V).(e
/I)>=h
/4π,
Grouping V and I together will give R and the quantity, 4π, since dimensionless can be grouped with R, and thus giving R'.
Therefore, R'=h
/2e
2.
Inverting this will give the conductance quantum.
Note: The above treatment was considered for understanding the physical limitation of conductance. The account may not be accurate mathematically, even though, it is possible physically.
References
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
2/h
Planck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...
= ≈ Ω−1. It appears when measuring the conductance of a quantum point contact
Quantum point contact
A Quantum Point Contact is a narrow constriction between two wide electrically conducting regions, of a width comparable to the electronic wavelength . Quantum point contacts were first reported in 1988 by a Dutch group and, independently, by a British group...
.
The name conductance quantum is somewhat misleading, since it implies that there can only be conductances that are integer multiples of G0. This is not the case, instead conductance quantum means the conductance of one conductance channel, if the probability of an electron entering this channel being transmitted is unity. If this probability is not unity, there must be a correction for the particular conductance channel.
An Intuitive Derivation from "uncertainty principleUncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
"
The Quantum of Conductance can easily be derived and seen to be as a consequence of the time-energy uncertainty.Let us assume that a wire is carrying a current I. The current is given by e
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
/τ, where τ is transit time and the e is electronic charge. When a potential of V is applied, then,
E=e
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
V and I= e
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
/τ, which implies τ=e
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
/I,
then, (e
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
V).(e
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
/I)>=h
Planck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...
/4π,
Grouping V and I together will give R and the quantity, 4π, since dimensionless can be grouped with R, and thus giving R'.
Therefore, R'=h
Planck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...
/2e
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
2.
Inverting this will give the conductance quantum.
Note: The above treatment was considered for understanding the physical limitation of conductance. The account may not be accurate mathematically, even though, it is possible physically.
References